- ELF.SEQ.E Exponential Equations

- ELF.SEQ.L Logarithmic Equations

Consider the function $f(x) = Ab^{kx}$.

A standard form of an exponential equation with variable $x$ is an equation of the form $f(x) = Ab^{kx} = C$ with $A, k \ne 0$.

Intermediate algebra spends a considerable amount of time solving this equation without any reference to functions.

When $k = 1$ this equation has the solution $x = \log_b( \frac C A)$.

When $k \ne 1$ the equation can be solved so that $x = \frac 1k \log_b( \frac C A)$.

A slightly more ambitious form of an exponential equation with variable $x$ is an equation of the form $ A_1b^{k_1x} = A_2b^{k_2x} $ with $A_1 \ne A_2$ and $k_1 \ne k_2$. Algebraically this is solved by solving the related equation $ Ab^{kx} = 1 $ where $A =\frac {A_1}{A_2}$ and $k = k_1 - k_2$.

Consider the function $f(x) = A\log_b(kx)$.

A standard form of a logarithmic equation with variable $x$ is an equation of the form $f(x) = A\log_b(kx)= C$ with $A, k \ne 0$.

Intermediate algebra spends a considerable amount of time solving this equation without any reference to functions.

When $k = 1$ this equation has the solution $x = b^{ \frac C A}$.

When $k \ne 1$ the equation can be solved so that $x = \frac 1k b^{ \frac C A}$.